An Innovative Control Design Procedure for Under-Actuated Mechanical Systems: Emphasizing Potential Energy Shaping and Structural Preservation

In this article, we propose a procedure to solve the controlled design for a class of under-actuated mechanical systems. Our proposed method can be viewed as a sub-method of the IDA-PBC or Controlled Lagrangian approaches, with a particular focus on shaping the potential energy. By emphasizing potential energy shaping, we can effectively tackle the bottleneck presented by the matching equation in these approaches. Moreover, our method leverages a suitable coordinate transformation that is inspired by the physics of the system, further enhancing its efficacy. Therefore, our design procedure is based on a coordinate transformation plus potential energy shaping in the new coordinates, and its existence and possibility of potential energy shaping can be verified via some algebraic calculations, making it constructive. To illustrate the results, we consider the cart-pole system and a recently introduced under-actuated mechanical system named swash mass pendulum (SMP) (Salamat and Tonello, 2021). The SMP consists of a pendulum made of a rigid shaft connected to a pair of cross-shafts where two swash masses can move under the action of servo-mechanisms.